SC PHIL 110 - Powerpoint for lesson 27 - D36195

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SC PHIL 110 - Powerpoint for lesson 27

School name University Of South Carolina-Columbia

Course Phil 110- Intro to Logic I

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7The Four Aristotelian FormsAll P’s are Q’s x (P(x) ∀  Q(x))Note the conditional. Some P’s are Q’s Ǝx (P(x) ᴧ Q(x))Note the conjunction. And note that this does NOT equal Ǝx (P(x)  Q(x))No P’s are Q’s x (P(x) ∀  ¬Q(x)) ¬Ǝx (P(x) ᴧ Q(x)) Some P’s are not Q’s Ǝx (P(x) ᴧ ¬Q(x))Twotipsinthe“Remember”boxonp.247:1) Translationsofcomplex quantified noun phrasesusuallyincludeconjunctionsofatomicpredicates. “A small, happy dog is at home.”Ǝx[(Small(x) ᴧ Happy(x) ᴧ Dog(x)) ᴧ Home(x)] (“Forsomex,xissmallandhappyandadogandathome”)“Every small dog that is at home is happy.”∀x [(Small(x) ᴧ Dog(x) ᴧ Home(x))  Happy(x)](“Forevery/eachx,ifxissmallandadogandathome,thenitishappy”)2) TheorderofanEnglishsentenceinvolvingcomplexnounphrasesdoesn’t always correspond totheorderofitsFOLtranslation.“Max owns a small, happy dog.”Ǝx[(Small(x) ᴧ Happy(x) ᴧ Dog(x)) ᴧ Owns(max, x)](“Forsomex,xissmallandhappyandadog,andMaxownsit”)Note that the complex noun phrase still comes first in FOL.“Max owns every small, happy dog.”∀x [(Small(x) ᴧ Happy(x) ᴧ Dog(x))  Owns(max,x)](“Forevery/eachx,ifxissmallandhappyandadog,thenMaxownsit”)POTENTIAL PROBLEM CASES:∀x (P(x)  Q(x))…invacuously truecases(i.e.,inworldswheretheantecedentisfalse,thusitisimpossibletofindacounterexample)e.g., x (Tet(x) ∀  Small(x))…thisisalwaysvacuouslytrueina world with no tetrahedra (andcanbeeithertrueorfalseinworldswithtetrahedra)Compare“EveryfreshmanwhotooktheclassgotanA”if no freshmen took the class! (inEnglishthissentenceinvolvesaconversationalimplicaturethatisabsentinFOL)e.g., x (Tet(x) ∀  Cube(x))…thisis‘inherently’ vacuous:it’strueonlywhenassertedofa world with no tetrahedra (andit’sfalseinallotherworlds)ANOTHER POTENTIAL PROBLEM CASE:“Some P’s are Q’s”doesNOTcontradict“All P’s are Q’s”inFOL…whynot?(Thisinvolvesanotherconversationalimplicature:InEnglish,whenweuse‘some’weusuallyimply‘notall’,butthisimplicatureisabsentfromFOL,soinFOLthereisnoupperboundon‘some’.)Quantifiers and function symbolsHavingquantifiersinourFOLallowsustoexpresssomeclaimsinvolvingfunctionsymbolsmuchmoreefficientlythanwecouldbefore.Forexample:∀x Nicer(father(father(x), father(x))“Foreveryx,thepaternalgrandfatherofxisnicerthanthefatherofx”Withoutfunctionsymbolswe’dhavetouseapredicatelikeFatherOftoexpressthesameidea,butwe’dneed3quantifiers:∀x y z ((FatherOf(x,y) ∀ ∀ ᴧ FatherOf(y,z))  Nicer(x,y))Thetextbookasksyoutodecidewhichofthefollowingsentencesistrueinallworldsandwhichistrueinonlysomeworlds:∀x (lm(lm(x)) = lm(x))∀x (fm(lm(x)) =

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